SCIENTIFIC NOTATION

Recall that *large* (or *big*) numbers are far away from zero, and
*small* numbers are close to zero.

Large and small numbers can be positive (to the right of zero) or negative (to the left of zero).

In this section, you'll do lots of work with powers of ten.

Recall that [beautiful math coming... please be patient] $\,10^3 = 1000\,$ and $\displaystyle\,10^{-3} = \frac{1}{10^3} = \frac{1}{1000}\,$.

More generally, [beautiful math coming... please be patient] $\,10^n\,$ is a ‘1’ followed by $\,n\,$ zeros; and $\,10^{-n}\,$ is $\,1\,$ over $\,10^n\,$.

Multiplying by
[beautiful math coming... please be patient]
$\,10^3\,$ is the same as multiplying by
$\,1000\,$.

Multiplying by ten to a *positive* power makes a number bigger.

Multiplying by
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$\,10^{-3}\,$ is the same as dividing by
$\,1000\,$.

Multiplying by ten to a *negative* power makes a number smaller.

Numbers that are very large (like
[beautiful math coming... please be patient]
$\,B = 23{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000\,$)

and very small (like
$\,S = 0{\bf.}00000000000000000000000000000000000000000000000006789\,$)

have long, inconvenient representations using standard notation.

*Scientific notation* gives a
representation for large and small numbers that is much more compact
and easier to work with.

Here are the scientific notation names for the big number
[beautiful math coming... please be patient]
$\,B\,$ and the small number
$\,S\,$ given above:

$B = 2.3 \times 10^{43}$

$S = 6.789 \times 10^{-50}$

Notice that there are two ‘parts’ to each number,
separated by the ‘times’ symbol,
‘$\times$’:

- the first part is a number between [beautiful math coming... please be patient] $\,1\,$ and $\,10\,$;
- the second part is a power of ten.

Notice that big numbers have ten raised to a positive power,
and small numbers have ten raised to a negative power.

Here are some more examples:

[beautiful math coming... please be patient] $3.1 \times 10^{-5}\,$ | is a small positive number (close to zero, to right of zero) |

$-3.1 \times 10^{-5}\,$ | is a small negative number (close to zero, to left of zero) |

$3.1 \times 10^{5}\,$ | is a big positive number (far from zero, to right of zero) |

$-3.1 \times 10^{5}\,$ | is a big negative number (far from zero, to left of zero) |

Usually, in algebra and beyond, the
‘$\,\times\,$’ symbol is *not* used for multiplication, because
it can too easily be confused with the variable $\,x\,$.

Scientific notation is the exception to this rule!

The precise definition of scientific notation follows.

Recall that the * integers* are the numbers:
[beautiful math coming... please be patient]
$\,\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$

DEFINITION
scientific notation

A number is in scientific notation if and only if it has the form
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$$d\times 10^n$$
where
$\,d\,$ is a number satisfying
$\,1 \le d < 10\,$ and
$\,n\,$ is an integer.

The number $\,d\,$ should be expressed as a decimal (as needed).

The number $\,d\,$ should be expressed as a decimal (as needed).

EXAMPLES:

[beautiful math coming... please be patient]
$1.027 \times 10^{-2}\,$ is in scientific notation.

Here, $\,d = 1.027\,$ is a number between $\,1\,$ and $\,10\,$, and $\,n = -2\,$ is an integer.

Here, $\,d = 1.027\,$ is a number between $\,1\,$ and $\,10\,$, and $\,n = -2\,$ is an integer.

[beautiful math coming... please be patient]
$7 \times 10^{1024}\,$ is in scientific notation.

Here, $\,d = 7\,$ is a number between $\,1\,$ and $\,10\,$, and $\,n = 1024\,$ is an integer.

Notice that no decimal point is needed in the number $\,7\,$.

Here, $\,d = 7\,$ is a number between $\,1\,$ and $\,10\,$, and $\,n = 1024\,$ is an integer.

Notice that no decimal point is needed in the number $\,7\,$.

[beautiful math coming... please be patient]
$10 \times 10^5\,$ is *not* in scientific notation,
because the first part must be strictly less than 10.

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$2 \times 10^{0.7}\,$ is *not* in scientific notation,
because $\,0.7\,$ is not an integer.

*Changing a number from scientific notation back to standard notation*
is a repeated application of the following simple rules:

- When you multiply by ten, you move the decimal point one place to the right.

Thus, multiplying by (say) [beautiful math coming... please be patient] $\,10^5\,$ moves the decimal point five places to the right. - When you divide by ten, you move the decimal point one place to the left.

Thus, multiplying by (say) [beautiful math coming... please be patient] $\,10^{-5}\,$, which is the same as dividing by $\,10^5\,$, moves the decimal point five places to the left.

EXAMPLES:

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$1.027 \times 10^{-2} = 0.0127$ (move decimal point two places to the left; it's good style to put the extra zero to the left of the decimal point)

[beautiful math coming... please be patient]
$7.1 \times 10^4 = 71,000$ (move decimal point four places to the right; insert appropriate commas for easy readability)

Numbers have lots of different names,
and one of the most common ways to rename a number is to multiply
by one in an appropriate form.

This idea is used to convert a number in standard notation to scientific notation:

you first ‘stretch or shrink’ to a number between $\,1\,$ and $\,10\,$,
and then restore to the original size, as illustrated next:

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$0.000037$

$= 0.000037 \times 1$ (Now, what name for $\,1\,$ shall we use?)

$= 0.000037 \times (10^5 \times 10^{-5})$ (Why this name for $\,1\,$? To turn $\,0.000037\,$ into $\,3.7\,$, move the decimal point $\,5\,$ places!)

$= (0.000037 \times 10^5) \times 10^{-5}$ (re-group)

$= 3.7 \times 10^{-5}\,$ (the idea: multiplying by $10^5$ *stretches* to get between $\,1\,$ and $\,10\,$;
multiplying by $\,10^{-5}\,$ *shrinks* back!)

People don't typically write out all these steps (phew)!

Instead, most people use the following ‘shortcut’ :

*Changing a number from standard notation to scientific notation:*

- Move the decimal point to obtain a number between [beautiful math coming... please be patient] $\,1\,$ and $\,10\,$, counting the number of places it is moved.
- If you moved it $\,n\,$ places to the right, then multiply by $\,10^{-n}\,$.
- If you moved it $\,n\,$ places to the left, then multiply by $\,10^n\,$.

EXAMPLES:

[beautiful math coming... please be patient]
$0.000037 = 3.7 \times 10^{-5}$ (move decimal point $\,5\,$ places to the right)

[beautiful math coming... please be patient]
$590.27 = 5.9027 \times 10^2$ (move decimal point $\,2\,$ places to the left)

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$39{,}000{,}000 = 3.9 \times 10^7$ (insert decimal point to right of last zero, move $\,7\,$ places to the left)

Master the ideas from this section

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Significant Figures and Related Concepts

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Significant Figures and Related Concepts

On this exercise, you will not key in your answer.

However, you can check to see if your answer is correct.

However, you can check to see if your answer is correct.